Magnetic properties of the transition to localized superconductivity around columnar defects

We consider flux lines in the presence of a dense square array of columnar defects, with insulating con and radius of the order of the coherence length. The properties of the phase transition from the flux line regime to the localized superconducting state are determined here. We show that the localized superconducting state is a new vortex state because of the counterclockwise and clockwise supercurrent flows found around each defect. We propose here a general operator to count the vorticity of any cavity inside a superconductor and give a method to obtain the reversible magnetization of superconductors with multiply connected geometries. [S0163-1829(99)11541-8].